arXiv subjects

Martin Ortiz

Publications and source records attributed to Martin Ortiz.

3 recordsLinked to original sources

arXiv2026

Theta operators on Hodge type Shimura varieties

We construct a new family of mod $p$ weight shifting differential operators on Hodge type Shimura varieties at hyperspecial level. First we construct basic theta operators, labelled by positive roots, that generalize Katz's theta operator for modular forms. Secondly we construct theta linkage maps, these are operators between automorphic vector bundles with linked weights, which can be thought of as generalizations of the classical theta cycle of Tate--Jochnowitz. In particular, there exist such maps within the $p$-restricted region, whose weight shifts are directly related to the conjectures of Herzig on the weight part of Serre's conjecture. We explain the relation between the two operators, and we prove some properties about them, e.g., the injectivity of some of them in a generic locus of the $p$-restricted region. As an application, we produce an example of a generic entailment of Serre weights for the groups $GL_{4,\mathbb{Q}_p}$ and $U(4)_{\mathbb{Q}_p}$, by combining the method of arxiv:2410.09602 with our stronger results about theta operators.

math.NT↗

arXiv2026

A de Rham weight part of Serre's conjecture and generalized mod $p$ BGG decompositions

We propose the use of de Rham cohomology of special fibers of Shimura varieties to formulate a geometric version of the weight part of Serre's conjecture. We conjecture that this formulation is equivalent to the one using Serre weights and the \'etale cohomology of Shimura varieties. We prove this equivalence for generic weights and generic non-Eisenstein eigensystems for a compact $U(2,1)$ Shimura variety such that $G_{\mathbb{Q}_p}=GL_3$. We do this by proving a generic concentration in middle degree of mod $p$ de Rham cohomology with coefficients. In turn, we prove this generic concentration by constructing generalized mod $p$ BGG decompositions for de Rham cohomology. After applying the results from our companion paper, this reduces to computing some BGG-like resolutions in a certain mod $p$ version of category $\mathcal{O}$, which is the main content of the article. In the $GSp_4$ case we also compute some explicit BGG decompositions, and assuming the generic concentration in middle degree of de Rham cohomology we obtain an improvement on the main result of arxiv:2410.09602.

math.NT↗

arXiv2024

Theta operators and a generic entailment for $GSp_4$

We construct a new family of mod $p$ weight shifting differential operators on the Siegel threefold. In particular, we construct one operator which generalizes the classical theta cycle, whose weight shift allows for maps between $p$-restricted weights, and which is generically injective on global sections. As an application we produce a generic entailment of Serre weights, i.e. any Hecke eigenform which is modular for a generic Serre weight in the lowest alcove is also modular for a Serre weight in one of the upper alcoves. The entailed Serre weight corresponds to a shadow weight of the lowest alcove Serre weight, in Herzig's conjectural description of $W(\overline{\rho})$.

math.NT↗